Recall from [1]} that if a finite algebra \(\mathbf {A}\) 
has a cube term, then for every subalgebra
\(\mathbf {B}\)  of any finite power \(\mathbf {A}^n\)  of \(\mathbf {A}\) 
there exists a sort of canonical set of generators,
called a compact representation in
[2]}, [1]}, [4]}
or a frame in [5]}.
Compact representations played
a crucial role in the CSP applications of finite algebras \(\mathbf {A}\)  with
cube terms (see [4]}).
For the subpower membership problem \(\operatorname{\text{\sc SMP}}(\mathbf {A})\) , a natural
approach to answering the question
“Is \(b\)  in the subalgebra \(\mathbf {B}\)  of \(\mathbf {A}^n\)  generated by
\(a_1,\dots ,a_k\) ?” would be to first find a
compact representation for \(\mathbf {B}\) , and then use it to answer the question.
The main components of a compact representation for \(\mathbf {B}\) 
are witnesses for forks.
For a
coordinate position \(i\in \lbrace 1,\dots , n\rbrace \)  and
for two elements
\(c,d\in A\) , the
tuples \((c_1,\dots ,c_n),(d_1,\dots ,d_n)\)  in \(\mathbf {B}\)  are said to witness
that
the pair \((c,d)\)  is a fork in \(\mathbf {B}\)  in position \(i\) 
if \(c_1=d_1,\dots ,c_{i-1}=d_{i-1}\)  and \(c_i=c,d_i=d\) .
It turns out
(see, e.g., Theorem REF  and
the paragraph preceding Lemma REF )
that in order to
give a positive answer to Question 1 it would be sufficient
to be able to decide in polynomial time
— using only the specified generators \(a_1,\dots ,a_k\)  —
whether or not a given pair \((c,d)\in A^2\)  is a fork in \(\mathbf {B}\)  in
a given position \(i\)  (and if it is, find witnesses for it in \(\mathbf {B}\) ).
This, however, appears to be a significant problem;
cf. Theorems REF , REF ,
and REF .
